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Let X(1)≤X(2)≤...≤X(2m+1) be an ordered sample of a random variable X, which has a median μ and a continuous probability distribution function. We consider the statistic Sm,r=(X(m+1)−μ)/(X(m+1+r)−X(m+1−r)), for some integer r, 1≤r≤m. This statistic is independent of the scale parameter and has other properties similar to Student’s t, and can be computed when the values of X(m+1), X(m+1+r), X(m+1−r) are known while some or all other sample values are not available (e.g. are censored out). This paper presents tables of exact critical values of Sm,r for small sample sizes, under the assumption that X has a normal probability distribution. Furthermore, a table is also given of asymptotic critical values for Sm,r, which can be used for large sample sizes even when X is not normally distributed. These tables may make possible the practical use of Sm,r in some situations in which Student’s t cannot be used.

Let $F(x)$ be the continuous distribution function of a random variable $X,$ and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$. It is well known that the probability $P_n(\epsilon)$ of $F(x)$ being everywhere majorized by $F_n(x) + \epsilon$ is independent of $F(x)$. The present paper contains the derivation of an explicit expression for $P_n(\epsilon)$, and a tabulation of the 10%, 5%, 1%, and 0.1% points of $P_n(\epsilon)$ for $n =$ 5, 8, 10, 20, 40, 50. For $n =$ 50 these values agree closely with those obtained from an asymptotic expression due to N. Smirnov.

Let X(1) ≤ X (2) ≤ … ≤ X (n) be an ordered sample of a random variable X which has continuous probability distribution function F (x), and let Fn (x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Renyi, are considered: The paper presents table of exact probabilities for these statistics for finite sample sizes. The limiting distributions of these statistics for sample size n ∞ are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions. Numerical examples for the use of the tables are presented, as well as applications to testing hypotheses on life distributions and to one-sided estimation of probability distribution functions from censored data.

A number of recent publications have dealt with problems of analyzing the performance of multi-component systems and evaluating their reliability. For example, a comprehensive theory of two-terminal networks was presented in [I] by Moore and Shannon who, among other results, have developed methods for obtaining highly reliable systems using components of low reliability; some of their procedures are credited to earlier work by von Neumann [2]. Several of the concepts and results of the present paper are generalizations of the corresponding concepts and results of the Moore-Shannon paper. A discussion of complex systems interpreted as Boolean functions may be found in the paper [3] by Mine. The present study deals with general classes of systems which contain two-terminal networks and most other kinds of systems considered previously as special cases, and investigates their combinatorial properties and their reliability. These classes consist, with several variants, of systems such that the more components...

SummaryA new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

nential life distributions are accepted as characterizing the phenomenon of no wear. The problem of finding a class of life distributions which would similarly reflect the phenomenon of wear-out has been under investigation for some time. In answer to this problem we introduce in this paper the class of IHRA (Increasing Hazard Rate Average) distributions and show that it has, among others, the following optimal properties: (i) it contains the limiting case of no wear, i.e., all exponential distributions, (ii) whenever components which have IHRA life distributions are put together into a coherent system, this system again has an IHRA life distribution, i.e., a system wears out when its components wear out, and (iii) the IHRA class is the smallest class with properties (i) and (ii). 1. Introduction. The nature of many physical devices is to wear out in time. In this paper we endeavor to determine how the wear-out process is reflected as a property of the corresponding life distributions. Wear-out is an intuitively suggestive concept, but its precise stochastic meaning is not immediately evident. However, as a beginning, we review the long accepted stochastic characterization of the phenomenon of no wear. Let T ? 0 be the failure time of a device. Let F(t) = PI T > t} be the complement of the usual distribution function, which we will call the "survival probability." The conditional survival probability for remaining life, given that the device has survived to age x, is Fx(t) = F(x + t)/F(x) if F(x) > 0, Fx(t) = 0 if F(x) = O. A device does not wear if regardless of age an unfailed device is like new, i.e. Fx(t) = Fo(t) for all x, t > 0. It follows that the class of no wear survival probabilities is the class of solutions of the functional equation

Miner’s rule for the cumulative damage due to fatigue, a deterministic formula which is well known in engineering practice, was examined earlier from a probabilistic point of view with Birnbaum in [2]. Here the assumptions of that model are weakened. Previously the basic assumptions were that crack growth was stochastic in nature with incremental extensions having a distribution with increasing failure rate, and that the spectrum of load fluctuations was fixed and then repeated under program. We now assume, instead of the distributions of incremental crack extension having increasing failure rate, only that for a given load fluctuation, the expected residual damage increment either in crack initiation or extension, given the damage exceeds a preassigned amount, is less than the damage increment which was expected for that load fluctuation before it was imposed. We also weaken the assumptions concerning the type of loading spectra which are admitted, considering the case of random load fluctuations which a...

Let $X, Y$ be independent random variables with continuous cumulative probability functions and let $$p = \mathrm{Pr}\{Y < X\}.$$ For the variance of the Mann-Whitney statistic $U,$ upper and lower bounds are obtained in terms of $p$, for the case of any $X$ and $Y$ as well as for the case of stochastically comparable $X, Y$. The results for the case of stochastic comparability are new, while the inequalities in the case of arbitrary $X, Y$ have either been obtained by van Dantzig or are a consequence of other inequalities due to van Dantzig.

The concept of a “module”, i.e., a package of components of a system which can be removed and replaced as a whole, has been long in use in systems design and analysis. In this paper a formal definition of this concept is given and its properties are studied.Most of the results are obtained for “coherent” systems, i.e., for systems whose performance improves as the performance of their components improves. For such systems the relationship between modules and minimal paths can be fully clarified, and the results obtained lead to a criterion for deciding whether or not a given set of components constitutes a module of a given system. It is shown that a coherent system always has uniquely determined maximal modules which have the property that either all are disjoint or no two of them are disjoint. These maximal modules determine uniquely a decomposition of the system into disjoint modular factors. Furthermore, a theorem on the union of modules (the “three modules theorem”), which has been previously known f...

A statistical model for life-lengths of structures under dynamic loading is derived. The model makes it possible to express the probability distribution of life-lengths in terms of the load given as a function of time and of deterioration occurring in time independently of loading. The special case of a constant load (or of periodic loading with constant amplitude) leads to gamma distributions for life lengths.

The concepts of distribution-free statistics and statistics of structure (d) were defined relative to families Omega and Omega' of cumulative probability functions. A counter example was presented to disprove the conjecture that every distribution-free statistic, symmetric in X1, X2...., Xn (a sample of a 1-dimensional random variable X with a continuous cumulative probability function F) with Omega = Omega' = Omega 2, has structure (d). Strongly distribution-free statistics in a family Omega* with respect to Omega' were defined, and a proof was presented of the equivalence of statistics of structure (d) and strongly distribution-free statistics under the condition that Omega = Omega' = Omega*.

A technique is presented for the treatment of errors introduced into sampling surveys due to the non-availability of respondents. The expected cost and variance of the sample survey are expressed as functions of sample size and of the number of call-backs made on the non-availables. A method is then presented which optimizes precision for a given cost by playing sampling error against the bias resulting from non-availables. * Research under the sponsorship of the Office of Naval Research. Presented to the Institute of Mathematical Statistics, November 27, 1948.

SummaryThe estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

Let $X_1 < X_2 < \cdots < X_n$ be a sample of size $n$, ordered increasingly, of a one-dimensional random variable $X$ which has the continuous cumulative distribution function $F$. It is well known, [1], that the statistic \begin{equation*}\tag{1}D^+_n = \sup_{-\infty < x < + \infty} \{F_n(x) - F(x)\},\end{equation*} where $F_n(x)$ is the empirical distribution function determined by $X_1, X_2, \cdots, X_n$, has a probability distribution independent of $F$. One may, therefore, assume that $X$ has the uniform distribution in (0, 1) and, observing that the supremum in (1) must be attained at one of the sample points, write without loss of generality \begin{equation*}\tag{2}D^+_n = \max_{1 \leqq i \leqq n} (i/n - U_i),\end{equation*} where $U_1 < U_2 < \cdots < U_n$ is an ordered sample of a random variable with uniform distribution in (0, 1). For a given $n > 0$ define the random variable $i^{\ast}$ as that value of $i$, determined uniquely with probability 1, for which the maximum in (2) is reached, i.e., such that \begin{equation*}\tag{3}D^+_n = i^{\ast}/n - U_{i^{\ast}},\end{equation*} and write \begin{equation*}\tag{3.1} U_{i^{\ast}} = U^{\ast}.\end{equation*} The main object of this paper is to obtain the distribution functions of $(i^{\ast}, U^{\ast})$, of $i^{\ast}$ and of $U^{\ast}$. The asymptotic distribution of $\alpha_n = i^{\ast}/n$ is also investigated, and bounds are obtained on the difference between the exact and the asymptotic distribution. A number of general identities, which are not commonly known, have been verified and used in proving the above-mentioned results. Since these identities may be helpful in other problems of this type, they are separated from the main proofs and appear in the next section.

0. Summary. In this paper we obtain some multivariate generalizations of Chebyshev's inequality, two of which are extended to continuous parameter stochastic processes. The extensions are obtained in a natural way by taking into account separability and letting the number of variables approach infinity. Particular attention is paid to the question of sharpness. To show that the bound of the inequality cannot be improved, examples are given in a number of cases that attain equality. 1. Introduction. We begin by discussing a model for the various generalizations of Chebyshev's inequality, and for a standard proof that we shall use. Examination of this proof will enable us to make some general comments concerning the problems of deriving inequalities and of proving sharpness. Let (Q, B, P) be a probability space, and let ($C, (1) be a measurable space. For each i c I, an arbitrary index set, let Bi C (t and let 5i be a class of random

Samples are often obtained under circumstances which make it less likely to draw individuals from some parts of a population than from others. In this paper a method is presented which makes it possible, under certain assumptions, to correct for the resulting bias and thus reconstruct the means, standard deviations, and correlation coefficients of the original population.

If $F(x)$ is a continuous distribution function of a random variable $X$, and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$, then the probability $\mathrm{Pr}\{F(x) \leqq F_n(x) + \epsilon \text{for all} x\}$ is known [1] to be a function $P_n(\epsilon)$, independent of $F(x)$. A closed expression for $P_n(\epsilon)$ and a table of some of its values were presented in [2]. In the present paper $P_n(\epsilon)$ is used to test a hypothesis $F(x) = H(x)$ against an alternative $F(x) = G(x)$. The power of this test is studied and sharp upper and lower bounds for it are obtained for alternatives such that $\sup_{-\infty < x < + \infty}\{H(x) - G(x)\} = \delta$, with preassigned $\delta$. The results of [2] are assumed known.

Let \begin{equation*}\tag{1.1}X^{(i)}_1,X^{(i)}_2, \cdots, X^{(i)}_{n_i},\qquad i = 1, 2, \cdots, c,\end{equation*} be samples of $c$ independent random variables $X^{(i)}$ with continuous cumulative distribution functions $F^{(i)}$, and let \begin{equation*}\begin{align*}F^{\ast^{(i)}}(x) &= 0\qquad x &< X^{(i)}_1 \\ \tag{1.2}F^{\ast^{(i)}} (x) &= k/n_i\qquad X^{(1)}_k &\leqq x < X^{(1)}_{k+1}, 1 \leqq k < n_i \\ F^{\ast^{(i)}} (x) &= 1\qquad X^{(i)}_{ni} \leqq x\end{align*}\end{equation*} be the corresponding $c$ emprirical distribution functions. We define the statistics \begin{equation*}\tag{1.3} D(n_1, n_2, \cdots, n_c) = \sup_{\substack{x, i, j\\(i,j=1,2,\cdots,c)}} |F^{\ast(i)} (x) - F^{\ast(j)} (x)|\end{equation*} and \begin{equation*}\tag{1.4} D^+(n_1, n_2, \cdots, n_c) = \sup_{\substack{x,i,j\\(i r\rbrack \\ P\lbrack D(n, n, \cdots, n) \leqq r\rbrack \geqq 1 - \lbrack c(c - 1)(c - 2)/6\rbrack P\lbrack D(n, n, n) > r\lbrack\end{align*}\end{equation*} are noted, which may be useful for values of $c \geqq 4$ for which tables are not available.